The notion apparently is of being "puffed up" with pride; compare Old English belgan "to become angry, offend, provoke," belg "anger, arrogance," from the same root as bellows and belly (n.). Meaning "a cause of boasting, occasion of pride" is from 1590s.Related: Boasted; boasting. An Old English word for "boasting" was micelsprecende "big talk."

boast (v.)

mid-14c., "to brag, speak arrogantly," from Anglo-French, from the same source as boast (n.). Meaning "speak with pride" is late 14c. Sense of "glory or exult in possessing" (something) is from 1540s; that of "possess something remarkable or admirable" is from 1690s. Related: Boasted; boasting.